Back to course
3.2PureStretch

Quadratic Simultaneous Equations

When one equation is linear and the other is quadratic, substitution is the way in. You end up with a single quadratic to solve, giving up to two solution pairs — the points where a line meets a curve.

30 min Equations and Inequalities
Edexcel AS level Maths: 3.2 Quadratic Simultaneous EquationsCurated by Kepler, not produced by us. Watch it before the notes below.
Open on YouTube

What you'll be able to do

  • Solve a linear–quadratic pair by substitution
  • Obtain and solve the resulting quadratic
  • Find both coordinates of each solution
  • Link the number of solutions to line–curve intersections
1

Always substitute the linear equation

Rearrange the linear\textbf{linear} equation to make one variable the subject, then substitute it into the quadratic. This produces a single quadratic in one variable.

Example 1Solve y=x+1y = x + 1 and x2+y2=25x^2 + y^2 = 25.
1Substitute y=x+1y = x + 1: x2+(x+1)2=25x^2 + (x + 1)^2 = 25.
2Expand: x2+x2+2x+1=25⇒2x2+2x−24=0x^2 + x^2 + 2x + 1 = 25 \Rightarrow 2x^2 + 2x - 24 = 0.
3Divide by 2 and factorise: x2+x−12=(x+4)(x−3)=0x^2 + x - 12 = (x + 4)(x - 3) = 0.
4So x=−4x = -4 or x=3x = 3; then y=x+1y = x + 1 gives y=−3y = -3 or y=4y = 4.
Answer
(−4,−3)(-4, -3) and (3,4)(3, 4)

Tip — Substitute the linear into the quadratic — never the other way round.

2

Pairing up the answers

Each xx-value has its own matching yy-value. Use the linear equation (the simplest one) to find each yy, and keep the pairs together.

Tip — Two x-values give two coordinate pairs. Do not mix them up.

3

Line meets curve

These solutions are the intersection points of a line and a curve. Two solutions means the line is a chord; one (repeated) means it is a tangent; none means it misses the curve entirely — exactly the discriminant idea again.

Formula recap

y=mx+c into y=ax2+bx+cy = mx + c \text{ into } y = ax^2 + bx + c
Substitute linear into quadratic.
2 / 1 / 0 solutions\text{2 / 1 / 0 solutions}
Chord / tangent / miss.

Common mistakes to avoid

Pairing every x with every y, giving four “solutions”.
Each x has one matching y from the linear equation — keep the pairs linked.
Substituting the quadratic into the linear.
Always substitute the linear equation into the quadratic.

Key takeaways

  • Make a variable the subject of the linear equation and substitute into the quadratic.
  • Solve the resulting single quadratic.
  • Find the matching second coordinate for each solution.
  • Solutions are line–curve intersections (chord/tangent/miss).

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

Browse practice topics

Related lessons

All 7 lessons

Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.